I caution against expecting strong resemblance between biological and artificial neural networks. I think the name "neural networks" is a bit dangerous, because it tricks people into expecting that neurological processes and machine learning should be the same. The differences between biological and artificial neural networks outweigh the similarities.
As an example of how this can go awry, you can also turn the reasoning in the original post on its head. You can train a neural network to learn to recognize cars in an afternoon, provided you have a reasonably fast computer and some amount of training data. You can make this a binary task (car/not car) or a multi-class task (car/tram/bike/airplane/boat) and still be confident in a high level of success.
By contrast, I wouldn't expect a child to be able to pick out a car the day - or even the week - after it's born, even after it has seen "so many training examples." Something is obviously different between a two-year-old and an infant that accounts for the difference in learning ability, whereas a vanilla image classification neural network is perfectly capable of picking up object classification immediately after "birth." I think that there are two important differences: (1) the relative volumes of training data available and (2) a self-teaching mechanism that develops over time because of abundant training data.
The original post exposes two questions. The title and body of the question ask why neural networks need "so many examples." Relative to a child's experience, neural networks trained using common image benchmarks have comparatively little data.
I will re-phrases the question in the title to
"How does training a neural network for a common image benchmark compare & contrast to the learning experience of a child?"
For the sake of comparison I'll consider the CIFAR-10 data because it is a common image benchmark. The labeled portion is composed of 10 classes of images with 6000 images per class. Each image is 32x32 pixels. If you somehow stacked the labeled images from CIFAR-10 and made a standard 48 fps video, you'd have about 20 minutes of footage.
A child of 2 years who observes the world for 12 hours daily has roughly 263000 minutes (more than 4000 hours) of direct observations of the world, including feedback from adults (labels). (These are just ballpark figures -- I don't know how many minutes a typical two-year-old has spent observing the world.) Moreover, the child will have exposure to many, many objects beyond the 10 classes that comprise CIFAR-10.
So there are a few things at play. One is that the child has exposure to more data overall and a more diverse source of data than the CIFAR-10 model has. Data diversity and data volume are well-recognized as pre-requisites for robust models in general. In this light, it doesn't seem surprising that a neural network is worse at this task than the child, because a neural network trained on CIFAR-10 is positively starved for training data compared to the two-year-old. The image resolution available to a child is better than the 32x32 CIFAR-10 images, so the child is able to learn information about the fine details of objects.
The CIFAR-10 to two-year-old comparison is not perfect because the CIFAR-10 model will likely be trained with multiple passes over the same static images, while the child will see, using binocular vision, how objects are arranged in a three-dimensional world while moving about and with different lighting conditions and perspectives on the same objects.
The anecdote about OP's child implies a second question,
"How can neural networks become self-teaching?"
A child is endowed with some talent for self-teaching, so that new categories of objects can be added over time without having to start over from scratch.
OP's remark about transfer-learning names one kind of model adaptation in the machine learning context.
In comments, other users have pointed out that one- and few-shot learning* is another machine learning research area.
Additionally, reinforcement-learning addresses self-teaching models from a different perspective, essentially allowing robots to undertake trial-and-error experimentation to find optimal strategies for solving specific problems (e.g. playing chess).
It's probably true that all three of these machine learning paradigms are germane to improving how machines adapt to new computer vision tasks. Quickly adapting machine learning models to new tasks is an active area of research. However, because the practical goals of these projects (identify new instances of malware, recognize imposters in passport photos, index the internet) and criteria for success differ from the goals of a child learning about the world, and the fact that one is done in a computer using math and the other is done in organic material using chemistry, direct comparisons between the two will remain fraught.
As an aside, it would be interesting to study how to flip the CIFAR-10 problem around and train a neural network to recognize 6000 objects from 10 examples of each. But even this wouldn't be a fair comparison to 2-year-old, because there would still be a large discrepancy in the total volume, diversity and resolution of the training data.
*We don't presently have a tags for one-shot learning or few-shot learning.